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I am looking for language characteristics only found in one single language.
(Maybe that could shed some lights on language development, since that could be a starting point to investegate why these characteristics are special. And further on, why / why not this characteristic would spread.)
Any language characteristics you believe is found in one language only, is much appreciated.
What language characteristics are unique to only one language?
If you want to find features attested in only one language, you could browse WALS. For example, when combining word order with alignment, there are many combinations that appear only once. However, restricting yourself to features that occur in exactly one language is unnecessary.
A feature being documented in only one language means neither that there are no undocumented or possible languages that share this feature. Hence one doesn't look at a feature occuring in only exactly one language but rather whether there are significantly more/less languages with the feature than without.
For example, the subject comes in most languages before the verb, whereas the object equally appears on either side. The fact that SV is far more common than VS is already significant, even though there is more than one language showing VS order.
There is much discussion about why SV is more common. It is assumed that some linguistic structures are easier for the human brain to produce or parse. Hence structures that are easier on the human brain would preferentially be innovated by languages.
However, the extant languages are not a statistically representative sample: many features we observe might also be because the communities of some language families out-bred and out-conquered other language families.
For example, while there are quite a few ergative languages in the world (especially clustering in Australia), Basque is an ergative language isolate in Europe (WALS tells you about geographical distribution, too).
While linguistic features are usually innovated within a language and then handed down to its descendants (unless they replace it by their own change), occasionally there's also borrowing through language contact - though that is mostly lexicon and far less grammar or phonology.
Hence when making conclusions you must be careful whether the featural distribution is through extralinguistic happenstance, due to being repeatedly innovated because it's easier to produce, due to language contact or due to any other kind of factor.
If you haven't already, I recommend reading some books relevant to this topic, e.g. Language Universals and Linguistic Typology by Bernard Comrie or Typology and Universals by William Croft or some books on historical linguistics.
If you are asking whether there are any features which in principle could only exist in a single language, there would be no such thing. OTOH if you are asking whether there are properties that have only been observed in a single language (living or dead), there are such things. For example, Nyole is the only language known to have the phonological rule ŋ → p / N ___. Chori is the only language with 6 contrastive surface tone levels. Dahalo is the only language with both clicks and pharyngeals (or epiglottals).
Note, by the way, that a WALS feature list is about the sampled languages. As it happens, that source identifies Maxakali as the only language with no fricatives of nasals, which seems to be correct since the other cluster of languages (Lushootseed, Twana, Makah, Quillayute) with no nasals has plenty of fricatives. It lists Lango as the only language with single exponence of TAM, agreement and "construct' (presence of an NP argument) -- it doesn't list Acholi, since Acholi is not part of the sample for this feature.
